7.3 Independent, Dependent and Inconsistent Systems

Independent and Dependent Systems

An independent system of equations is a system with a unique solution. All the examples and exercises in 7.2 were examples of independent systems.

Consider the following system:

\( \mathbb{S}_1\)

\(\begin{eqnarray*}
x + y + z &=& 1\\
y + z &=& 2
\end{eqnarray*}\)

Since there are fewer equations in the system than variables, there will not be a unique solution. In order to characterize the solutions of this system, we add a third equation to represent all possible values of \(z\):